Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 7 x^{2} - 36 x^{3} + 81 x^{4}$ |
| Frobenius angles: | $\pm0.0656128605321$, $\pm0.601053806135$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $49$ | $6321$ | $470596$ | $42028329$ | $3500716849$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $80$ | $642$ | $6404$ | $59286$ | $530486$ | $4778934$ | $43058564$ | $387442578$ | $3486722000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+x^5+(2a+1)x^4+2ax^3+(2a+2)x^2+2x+a$
- $y^2=(2a+1)x^6+2ax^5+(a+1)x^3+x^2+(2a+1)x+a$
- $y^2=ax^6+(a+2)x^5+(2a+2)x^3+(a+2)x^2+(2a+1)x+a$
- $y^2=ax^6+(a+2)x^5+(2a+1)x^4+(2a+1)x^3+ax^2+a$
- $y^2=(2a+1)x^6+(a+2)x^5+2ax^4+(2a+1)x^3+(a+2)x^2+ax+a+2$
- $y^2=ax^6+2ax^5+(a+2)x^4+ax^3+2ax^2+(2a+1)x+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-5})\). |
| The base change of $A$ to $\F_{3^{6}}$ is 1.729.abs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
Base change
This is a primitive isogeny class.